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Example Questions
Example Question #1 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let  be a simple closed contour, described positively. If a functionÂ
 is analytic insideÂ
 except for a finite number of singular pointsÂ
 insideÂ
, thenÂ
 Brown, J. W., & Churchill, R. V. (2009). Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use Cauchy's Residue Theorem to evaluate the integral of
 Â
in the region .
Note, forÂ
a singularity exists where . Thus, since where
 is the only singularity forÂ
 insideÂ
,  we seek to evaluate the residue forÂ
.
Observe,
The coefficient of  isÂ
.
Thus,Â
.
Therefore, by Cauchy's Residue Theorem,
Hence,
Â
Example Question #2 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let  be a simple closed contour, described positively. If a functionÂ
 is analytic insideÂ
 except for a finite number of singular pointsÂ
 insideÂ
, thenÂ
Brown, J. W., & Churchill, R. V. (2009). Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Using Cauchy's Residue Theorem, evaluate the integral of Â
Â
in the regionÂ
Note, forÂ
a singularity exists where . Thus, since whereÂ
 is the only singularity forÂ
 insideÂ
,  we seek to evaluate the residue forÂ
.
Observe,
The coefficient of  isÂ
.
Thus,Â
.
Therefore, by Cauchy's Residue Theorem,
Hence,
Â
Example Question #1 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let  be a simple closed contour, described positively. If a functionÂ
 is analytic insideÂ
 except for a finite number of singular pointsÂ
 insideÂ
, thenÂ
 Brown, J. W., & Churchill, R. V. (2009). Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use Cauchy's Residue Theorem to evaluate the integral of
 Â
in the region .
Note, there is one singularity for  whereÂ
.Â
LetÂ
Then
so
.
Therefore, there is one singularity for  whereÂ
. Hence, we seek to compute the residue forÂ
 whereÂ
Observe,
So, when ,Â
.
Thus, the coefficient of  isÂ
.
Therefore,Â
Hence, by Cauchy's Residue Theorem,
Therefore,
Example Question #1 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let  be a simple closed contour, described positively. If a functionÂ
 is analytic insideÂ
 except for a finite number of singular pointsÂ
 insideÂ
, thenÂ
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function  is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contourÂ
, then
Â
Brown, J. W., & Churchill, R. V. (2009). Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
 Â
in the region .
0
0
Note, Â
Thus, seeking to apply the Residue Theorem above for   insideÂ
, we evaluate the residue forÂ
.
Observe,
The coefficient of  isÂ
.
Thus,Â
.
Therefore, by the Residue Theorem above,
Hence,
Â
Example Question #1 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let  be a simple closed contour, described positively. If a functionÂ
 is analytic insideÂ
 except for a finite number of singular pointsÂ
 insideÂ
, thenÂ
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function  is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contourÂ
, then
Brown, J. W., & Churchill, R. V. (2009). Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
 Â
in the region .
Note, Â
Thus, seeking to apply the Residue Theorem above for   insideÂ
, we evaluate the residue forÂ
.
Observe,
The coefficient of  isÂ
.
Thus,Â
.
Therefore, by the Residue Theorem above,
Hence,
Â
Example Question #5 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let  be a simple closed contour, described positively. If a functionÂ
 is analytic insideÂ
 except for a finite number of singular pointsÂ
 insideÂ
, thenÂ
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function  is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contourÂ
, then
Â
Brown, J. W., & Churchill, R. V. (2009). Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
 Â
in the region .
Note, Â
Thus, seeking to apply the Residue Theorem above for   insideÂ
, we evaluate the residue forÂ
.
Observe,
The coefficient of  isÂ
.
Thus,Â
.
Therefore, by the Residue Theorem above,
Hence,
Â
Example Question #2 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let  be a simple closed contour, described positively. If a functionÂ
 is analytic insideÂ
 except for a finite number of singular pointsÂ
 insideÂ
, thenÂ
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function  is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contourÂ
, then
Brown, J. W., & Churchill, R. V. (2009). Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
 Â
in the region .
Note, Â
Thus, seeking to apply the Residue Theorem above for   insideÂ
, we evaluate the residue forÂ
.
Observe, the coefficient of  isÂ
.
Thus,Â
.
Therefore, by the Residue Theorem above,
Hence,
Â
Example Question #1 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let  be a simple closed contour, described positively. If a functionÂ
 is analytic insideÂ
 except for a finite number of singular pointsÂ
 insideÂ
, thenÂ
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function  is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contourÂ
, then
Brown, J. W., & Churchill, R. V. (2009). Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
 Â
in the region .
Note, Â
Thus, seeking to apply the Residue Theorem above for   insideÂ
, we evaluate the residue forÂ
.
Observe,
The coefficient of  isÂ
.
Thus,Â
.
Therefore, by the Residue Theorem above,
Hence,
Â
Example Question #8 : Residue Theory
Find the residue of the functionÂ
.
Observe
The coefficient of  isÂ
.
Thus,
.
Example Question #3 : Residue Theory
Find the residue at  ofÂ
.
Let .Â
Observe,
The coefficient of  isÂ
 since there is noÂ
 term in the sum.
Thus,Â
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